# Error analysis of rack and pinion steering mechanism

At present, the steering mechanism used in automobile can not fully meet the Ackerman steering principle, and can not realize the pure rolling of wheels in the steering process. Therefore, the error analysis of the steering mechanism has a positive guiding significance for the design and optimization of the steering mechanism. Rack and pinion steering mechanism is a common steering mechanism, which is mainly composed of rack and pinion steering gear, trapezoid arm, tie rod, etc. The torque drives the rack to move left and right through the gear in the steering gear, drives the tie rod and trapezoid arm to deflect the left and right steering wheels, and realizes the steering of the car [1]. According to the symmetry of the gear rack steering mechanism, the expressions of the angles α and β can be obtained from the geometric relationship,.

Left steering wheel angle

α=θ0-arccosk-m2+se2+ (k-m2+s) 2-arccosr2+ (e2+ (k-m2+s) 2) -[ (k-m2-rcosθ0) 2+ (rsinθ0-e) 2]2re2+ (k-m2+s) 2    (1)

Right steering wheel angle

β=arccosk-m2-se2+ (k-m2-s) 2+arccosr2+ (e2+ (k-m2-s) 2) -[ (k-m2-rcosθ0) 2+ (rsinθ0-e) 2]2re2+ (k-m2-s) 2-θ0    (2)

Four kinds of error analysis

#### 1. Steering wheel angle error

According to the Ackerman steering principle, the angle β of the right steering wheel can be obtained when the left steering wheel turns over α, which is called the ideal right steering wheel angle; according to the plane mathematical model, according to the relationship between the left and right steering wheel angles of the rack and pinion steering mechanism, the angle β ‘of the right steering wheel can also be obtained, which is called the actual right steering wheel angle [1]. The difference w between the ideal right steering wheel angle and the actual right steering wheel angle is called the steering wheel angle error [1].

w=|β-β′|    (3)

#### 2. Steering center error

As shown in Figure 2, when the angle of left and right steering wheels of the car cannot meet the Ackerman principle, the intersection o of the two steering wheel axes falls outside the rear wheel axis. Suppose that the intersection of the left and right steering wheel axes on the rear wheel axis is O1 and O2 respectively, at this time, the distance between o point and the rear wheel axis is defined as steering center error I, and the distance between the intersection of the left and right steering wheels on the rear wheel axis is defined as steering center error II [2]. According to the geometric relationship shown in Figure 3:

|ΟΗ|=kcotβ-cotα-l    (4) |Ο1Ο2|=l⋅ (cotα-cotβ) +k    (5)

The smaller the deviation between the actual steering center and the rear wheel axis is, the smaller the steering center errors I and II are, and the closer to the requirements of Ackerman steering principle, the better the steering characteristics of the steering mechanism are. Among them, steering center error I directly represents the deviation between steering center and rear wheel axis, which is more intuitive than steering center error II.

#### 3. Steering toe in error

In order to meet the Ackerman steering principle, it is assumed that the actual axis of the left and right steering wheels is shifted by Δ θ in the opposite direction, so that the axis of the left and right steering wheels and the axis of the rear wheels intersect at one point. This hypothetical deviation angle Δ θ is the steering toe angle [2]. The steering toe of the rack and pinion steering mechanism is shown in Figure 3, and the instantaneous steering center is o ‘. At this time, the left and right steering wheel angle and steering toe meet the relationship

cot (β+Δθ) -cot (α-Δθ) =kl    (6)

Solution (6) can be obtained

Δθ=arccot ( (2+2B+AC) ± (2+2B+AC) 2+4 (A-C) (A-BC) 2 (A-C) )     (7)

Where: a = cot β – cot α; b = cot β. Cot α; C = KL.

The toe in angle of automobile steering can represent the steering characteristics of automobile. When the vehicle is in over steering state, the steering toe is greater than zero; when the vehicle is in under steering state, the steering toe is less than zero [3]. When the actual turning angle of left and right steering wheels is closer to the intersection state, the steering mechanism is closer to the requirements of Ackerman steering principle, and the steering toe angle is smaller at this time.

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